Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Lojasiewicz Conditions
MCML Authors
Massimo Fornasier
Prof. Dr.
Principal Investigator
Abstract
Massimo Fornasier
Prof. Dr.
Principal Investigator
Abstract
In this work, we take a step towards understanding overdamped Langevin dynamics for the minimization of a general class of objective functions . We establish well-posedness and regularity of the law ρt of the process through novel a priori estimates, and, very importantly, we characterize the large-time behavior of ρt under truly minimal assumptions on . In the case of integrable Gibbs density, the law converges to the normalized Gibbs measure. In the non-integrable case, we prove that the law diffuses. The rate of convergence is (1/t). Under a Polyak-Lojasiewicz (PL) condition on , we also derive sharp exponential contractivity results toward the set of global minimizers. Combining these results we provide the first systematic convergence analysis of Langevin dynamics under PL conditions in non-integrable Gibbs settings: a first phase of exponential in time contraction toward the set of minimizers and then a large-time exploration over it with rate (1/t).
BibTeXKey: FSW25